A certain element has a half life of 4.5 billion years. 
a. You find a rock containing a mixture of the element and lead. You determine that 55​% of the original element​ remains; the other 45​% decayed into lead. How old is the​ rock?
b. Analysis of another rock shows that it contains 20​% of its original​ element; the other 80​% decayed into lead. How old is the​ rock?
Question content area bottom
Part 1
a. The rock is approximately 
  
3.9 billion years old. 
​(Round to one decimal place as​ needed.)
Part 2
b. The rock is approximately 
  
enter your response here billion years old. 
​(Round to one decimal place as​ needed.)

To solve this problem, we will use the formula for exponential decay, which is typically used for half-life problems. The formula is:

\[
N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T}}
\]

Where:
- \( N(t) \) is the remaining amount of the substance at time \( t \),
- \( N_0 \) is the initial amount of the substance,
- \( t \) is the time that has passed,
- \( T \) is the half-life of the substance.

Alternatively, we can express the proportion of the remaining element as:

\[
\frac{N(t)}{N_0} = \left( \frac{1}{2} \right)^{\frac{t}{T}}
\]

### Part 1:
The first rock has 55% of the original element remaining. This gives us:

\[
\frac{N(t)}{N_0} = 0.55
\]

The half-life \( T \) of the element is 4.5 billion years. We want to solve for \( t \) (the age of the rock):

\[
0.55 = \left( \frac{1}{2} \right)^{\frac{t}{4.5}}
\]

To solve for \( t \), take the natural logarithm (ln) of both sides:

\[
\ln(0.55) = \ln\left(\left( \frac{1}{2} \right)^{\frac{t}{4.5}}\right)
\]

Using logarithmic properties:

\[
\ln(0.55) = \frac{t}{4.5} \cdot \ln\left(\frac{1}{2}\right)
\]

Now solve for \( t \):

\[
t = 4.5 \cdot \frac{\ln(0.55)}{\ln(0.5)}
\]

Let me calculate that for you.

The age of the first rock is approximately \( 3.9 \) billion years.

### Part 2:
The second rock has 20% of the original element remaining, meaning:

\[
\frac{N(t)}{N_0} = 0.20
\]

Following the same process:

\[
0.20 = \left( \frac{1}{2} \right)^{\frac{t}{4.5}}
\]

Take the natural logarithm of both sides and solve for \( t \):

\[
t = 4.5 \cdot \frac{\ln(0.20)}{\ln(0.5)}
\]

I'll calculate this as well.The age of the second rock is approximately \( 10.4 \) billion years.

### Summary:
a. The first rock is approximately 3.9 billion years old.  
b. The second rock is approximately 10.4 billion years old.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. How does the amount of remaining element affect the age calculation?
2. What is the relationship between half-life and decay rate?
3. How would the calculations change if the half-life were different?
4. What happens to the decay formula if we know the fraction that decayed instead of the fraction remaining?
5. How accurate are these age estimates in real-world geological analysis?

**Tip:** When solving exponential decay problems, taking the natural logarithm (ln) is a crucial step for isolating the time variable.

A certain element has a half-life of 4.5 billion years. a. You find a rock containing a mixture of the element and lead. You determine that 55% of the original element remains; the other 45% decayed into lead. How old is the rock? b. Analysis of another rock shows that it contains 20% of its original element; the other 80% decayed into lead. How old is the rock?

Learn how to calculate the age of rocks using the half-life of elements and the exponential decay formula. In this example, we determine the age of rocks with 55% and 20% of the original element remaining, respectively, resulting in rock ages of 3.9 and 10.4 billion years.

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